论文标题
局部有限图上的对数Schrödinger方程的基态
Ground states for logarithmic Schrödinger equations on locally finite graphs
论文作者
论文摘要
In this paper, we study the following logarithmic Schrödinger equation \[ -Δu+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $Δ$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded从下方开始，可能会更改标志。在对$ a（x）$强加的不同假设时，我们首先建立了两个Sobolev紧凑型定理。它导致了两种相关的能量功能，其中一个在对数非线性下不是明确的，而另一种是$ c^1 $。然后，通过使用nehari歧管方法获得基态溶液的存在，并分别通过定理通过。
In this paper, we study the following logarithmic Schrödinger equation \[ -Δu+a(x)u=u\log u^2\ \ \ \ \mbox{in }V, \] where $Δ$ is the graph Laplacian, $G=(V,E)$ is a connected locally finite graph, the potential $a: V\to \mathbb{R}$ is bounded from below and may change sign. We first establish two Sobolev compact embedding theorems in the case when different assumptions are imposed on $a(x)$. It leads to two kinds of associated energy functionals, one of which is not well-defined under the logarithmic nonlinearity, while the other is $C^1$. The existence of ground state solutions are then obtained by using the Nehari manifold method and the mountain pass theorem respectively.
